I have been trying to understand and deconstruct the climate-modeling work of Philip Mulholland and Stephen Wilde (M&W). M&W seem to believe that the model they have developed explains planetary temperatures as a consequence of atmospheric mass movement, without any need to reference the radiative effects of greenhouse gases.

* * *

I want to talk about the paper Modelling the Climate of Noonworld: A New Look at Venus by Philip Mulholland and Stephen Wilde. This paper lays out a framework for planetary climate modeling which M&W call the Dynamic-Atmosphere Energy-Transport (DAET) model. In subsequent papers, M&W have applied their DAET model to Titan and Earth and to further study of Venus.

I’ve carefully examined the DAET model. My conclusions are:

- Unfortunately, the planetary temperature predicted by the DAET model relies on an inappropriate method of calculating temperature. This invalidates all the key claims of the work.
- Additionally, the DAET model isn’t rooted in physics, which means that there is little assurance that the predictions of the model will have much relationship to reality.

If you’d like details, read on.

* * *

“Noonworld” is the name M&W give to a hypothetical tidally-locked planet, with an atmosphere (nominally pure nitrogen) which is fully transparent to both shortwave and longwave radiation.

The Lit hemisphere always faces the Sun, and the Dark hemisphere is in perpetual darkness. The temperature difference between the Lit side and the Dark side induces convection in the atmosphere. The result is a circulatory system (somewhat like a Hadley Cell on Earth) that transports heat from the warmer Lit side to the cooler Dark side.

The planetary energy flow model is illustrated above in M&W’s Figure 4 (reflecting model values tuned to correspond to the insolation and observed temperature of Venus).

The model involves five energy flows:

- S
**“Solar Insolation”**(Insolation absorbed on the Lit side) - R₊
**“Lit Surface Radiant Loss to Space”**(Thermal radiation from the surface of the Lit hemisphere) - R₋
**“Dark Surface Radiant Loss to Space”**(Thermal radiation from the surface of the Dark hemisphere) - Aₓ
**“Top of Atmosphere Thermal Export”**(High-altitude warm air flow from the Lit side to the Dark side) - Aᵣ
**“Surface Cold Air Thermal Return”**(Low-altitude cool air flow from the Dark side to the Lit side)

I’ve used my own symbols to denote these flows. (M&W don’t provide any symbols, and I’d like to be able to express the mathematical relationships between the various quantities.)

M&W’s DAET model of Noonworld and Venus is rooted in a single premise:

- When an energy flow enters a hemisphere, it is
*partitioned between the surface radiant heat loss and the atmospheric energy flow leaving the hemisphere, according to a fixed ratio.*

Although M&W describe the energy partitioning in terms of a ratio, I’ll be describing it in an equivalent way, using an energy partition fraction, 𝛾. A fraction 𝛾 of the energy flow entering a hemisphere is assumed to enter (or continue in) the atmospheric circulation, and a fraction (1- 𝛾) is assumed to be radiated by the surface as thermal radiation.

**Mathematical Details**

*Those put off by math (or eager to get to the critique) are encouraged to skip to the next section, “Incompatible Temperatures.”*

W&M don’t unpack the technical meaning of the “energy flows” in their model. However, it seems clear what must be meant:

- “Solar insolation” and “radiant loss” energy flows technically refer to “irradiance” and “radiant exitance” (“radiant emittance”) values for received and emitted radiation. (The jargon for measuring radiation is very technical and specific, with many similar-sounding terms that have distinct meanings.) These values are averaged over a hemisphere of the planet.
- For consistency, the atmospheric “thermal export” and “thermal return” energy flows must refer to the total energy flow between hemispheres divided by the area of a hemisphere.

All energy flows are measured in units of watts per square meter. During many of their calculations, M&W further normalize energy flow values by expressing them relative to the average absorbed isolation, S.

The average insolation absorbed by the Lit hemisphere is given by S = (1-A₀)⋅S₀/2, where S₀ is solar irradiance at the relevant distance from the Sun and A₀ is the planetary albedo.

The energy-partitioning rule implies that energy flows are related as follows:

R₊ = (1 – 𝛾)⋅(S + Aᵣ)

Aₓ = 𝛾⋅(S + Aᵣ)

R₋ = (1 – 𝛾)⋅Aₓ

Aᵣ = 𝛾⋅Aₓ

Although M&W mainly present numerical solutions to the model, these equations can be solved analytically, yielding:

R₊ = S/(1+ 𝛾)

Aₓ = S⋅𝛾/(1−𝛾²)

R₋ = S⋅𝛾/(1+ 𝛾)

Aᵣ = S⋅𝛾²/(1−𝛾²)

If follows that the total thermal radiance is R₊+R₋ = S. It’s reassuring that the total thermal radiance is always S. That means the model conserves energy and guarantees radiant balance between the energy absorbed and emitted by the planet.

The net atmospheric heat flow from the Lit side to the Dark side is Aₓ−Aᵣ = S⋅𝛾/(1+𝛾). This means that, depending on 𝛾, up to half the absorbed solar flux may be transported from the Lit side to the Dark side. It’s reassuring that the heat transport is confined to that range.

* * *

M&W calculate what they call the Total Global Energy Budget (TGEB). This can be conceptualized two ways:

- The sum of all the thermal radiance and atmospheric energy flows, TGEB = R₊+R₋+Aₓ+Aᵣ
- The sum of the energy flows arriving at the Lit side, S+Aᵣ, and the energy flow arriving at the Dark side, Aₓ.

These two formulations are mathematically equivalent. Both yield TGEB = S/(1−𝛾).

As far as I can tell, W&M’s Total Global Energy Budget, TGEB, has no physical significance.

Perhaps M&W eventually came to realize this. When they reported the predictions of their model for 𝛾 = ½ (in Table 7), they reported TGEB and included a corresponding “temperature.” However, when they reported the predictions for a larger value of 𝛾 (in Table 9), they reported TGEB, but didn’t report any corresponding “temperature.” Perhaps they noticed that the result was difficult to rationalize as being a meaningful temperature?

* * *

G&W distinguish what they call “Diabatic” and “Adiabatic” versions of their model. The “Diabatic Model” sets the energy partition fraction to 𝛾 = ½, yielding TGEB = 2⋅S. The Adiabatic Model” involves an energy partition fraction 𝛾 > ½, yielding TGEB > 2⋅S.

I’m not certain, but I imagine the names relate to a belief that the model’s behavior for 𝛾 > ½ is a consequence of adiabatic processes in the atmosphere.

G&W apply the “Diabatic” model to Noonworld, and conclude that the “Adiabatic” model is needed to explain the temperature of Venus

To fit their model to Venus, G&W use a partition ratio 𝛾 = 0.991138. This yields atmospheric “thermal Export” and “thermal Return” energy flows respectively 56.17 and 55.67 times larger than the Lit hemisphere absorbed insolation, S = 299.15 W/m². Based on the Stephan-Boltzmann law, G&W calculate corresponding “temperatures” and assert that the global mean air temperature is 737 K (464℃), which matches their assumed mean temperature for Venus.

* * *

That’s M&W’s DAET model. Does it provide a useful model of how planetary temperatures are established?

I don’t think so. There is a “temperature” problem that invalidates the model, and there are other issues which further render the model suspect, as I’ll explain.

**Incompatible Temperatures**

Above is M&W’s Table 9, which shows energy flows and associated temperatures as predicted by their DAET model of Venus.

The “radiant loss to space” values for the Lit and Dark sides are calculated as corresponding to temperatures of -46.1℃ and -46.6℃, respectively. These temperatures are calculated using the Stephan-Boltzmann law, j* = σT⁴ where j* is the radiant exitance (radiant emittance).

This means that, in the model, the *surface of Venus* *has a temperature of about -46**℃**.*

M&W also apply the Stephan-Boltzmann law to the atmospheric “thermal export” and “thermal return” values, concluding that the “average global air temperature” is 464℃.

Compare those two temperatures. The model says the *surface* of Venus is at -46℃, but that the *atmosphere* is 500℃ hotter than the surface!

Do I need to say that that is not thermodynamically possible?

**Non-Physical “Temperature” Calculations**

As one can see from Table 9, M&W are happy to calculate a corresponding “temperature” for nearly every energy flow “Power Intensity Flux” in their model.

The problem with this is: the Stephan-Boltzmann law is only relevant to *thermal radiation!*

It is meaningless to apply it to any other sort of energy flow.

Suppose an audio speaker is playing music with an average acoustic energy flux of 1 watt/m². Applying the Stefan-Boltzmann (S-B) law to that acoustic energy flux yields a temperature of 65 K (-208℃).

Does anyone think such a calculation is meaningful?

I can hear the protests now: “But M&W are talking about convective air circulation carrying thermal energy, which relates to temperature, so surely it makes sense to apply S-B *there*, doesn’t it?”

No, it doesn’t.

What are the atmospheric “thermal export” and “thermal return” energy flows?

They don’t represent “heat flux” because heat can only flow from the hot Lit side to the cold Dark side, not in the other direction.

The energy fluxes are likely intended to represent the movement of the air’s total energy density, consisting of the sum of internal energy, U, and potential energy, PE.

So, the energy flux would be given by 𝚽 = (U + PE)⋅v where v is the velocity of air movement.

How does the energy flux 𝚽 relate to the temperature of the air, T?

Glossing over nuances to keep things simple, one could say U = C⋅T where C is heat capacity. Plugging this into the formula for energy flux and solving for T yields:

T = (𝚽/v – PE)/C

This formula has an interesting implication. The energy flows in the atmosphere increase as the partition fraction, 𝛾, is increased. Yet, this increase in energy flow *need not reflect any increase in air temperature*. A higher 𝛾 value might simply correspond to a higher value of atmospheric heat capacity, C, or circulation velocity, v.

Increasing either heat capacity or velocity would lead to a larger atmospheric energy flow and greater efficiency in transferring heat from the Lit hemisphere to the Dark hemisphere, without involving any increase in air temperature.

So, air temperature does have a relationship to the energy flows that M&W associate with circulation of the atmosphere. However, energy flux does not uniquely determine air temperature. And, calculating *the temperature of the air has* *nothing to do with the Stephan-Boltzmann law*.

* * *

Calculating a temperature from an energy flow via the Stephan-Boltzmann law is valid only if the energy flow refers to thermal radiation, or to a combination of energy flows that are logically known to be equal to the amount of thermal radiation. In addition, even when dealing with thermal radiation, one cannot simply add fluxes (rather than averaging them) and compute a temperature.

M&W seem to have no idea when it is or is not appropriate to apply the Stephan-Boltzmann law. As a result, most of the temperatures they calculate in Table 9 are nonsense.

In particular, all the air temperatures M&W compute are nonsense, the meaningless output of an inapplicable formula. Unfortunately, the “average global air temperature” is the central output of the DAET model, the result upon which M&W base all their conclusions.

The *only* physically meaningful temperatures that M&W compute in Table 9 are the ones saying that *the surface is at -46**℃**.*

Given that M&W are trying to explain a 464℃ near-surface temperature for Venus, this does not constitute a good fit between the model and reality.

* * *

My earlier review of a paper by Wilde and Mulholland revealed similar issues of temperature being calculated from energy flows in an inappropriate way.^{1}I didn’t realize at the time that that particular paper did not reveal the full essence of M&W’s approach to modeling. So, while my critique reflected what it was like to try to make sense of one of M&W’s papers in isolation, it didn’t address their full model. This appears to be an ongoing core flaw in M&W’s work.

* * *

This issue of inappropriate “temperature” calculations invalidates the predictions of the DAET model upon which M&W base their conclusions.

Given that, one might feel it is pointless to examine the model further. So, I’ll understand if you choose to stop reading at this point.

However, for completeness, I’ll comment on a few lesser issues.

**Peculiar Energy Flows**

Let’s look at the energy flows that M&W’s DAET model predicts.

The diagram above depicts the energy flows between the Sun, the Lit and Dark hemisphere surfaces, the “thermal export” and “thermal return” air currents, and space.

The numerical values are based on an energy partition ratio 𝛾 = 0.9. Energy is conserved, though numerical rounding might suggest small discrepancies.

In this system, the Lit hemisphere acts as the *heat source*, absorbing solar insolation. The Dark hemisphere acts as the *heat sink,* radiating energy to space.

Thermodynamically, the temperatures of the air in this system *must* be between the temperatures of the heat source and the heat sink.

In particular, it must be true that T₊ > Tₓ > Tᵣ > T₋ where T₊ is the temperature of the Lit hemisphere surface, Tₓ is the temperature of the “thermal export” air current (when at the altitude of the surface), Tᵣ is the temperature of the “thermal return” air current, and T₋ is the temperature of the Dark hemisphere surface.

I notice two peculiar things about the predicted energy flows:

- The energy flow from the Lit hemisphere surface to the air, 0.90, is much larger than the energy flow from the air to the Dark hemisphere surface, 0.47.

This is surprising. Thermodynamically, heat is flowing from the Lit surface to the air to the Dark surface, where it is then radiated. The heat flux is the same throughout this process.

So, one might expect that the amount of energy the surface transfers to the air on the Lit side would match the amount of energy the air transfers to the surface on the Dark side. - Energy flows at a non-zero rate, 0.43, from the
*cool*“thermal return” air flows to the*hot*surface of the Lit hemisphere.

We know that heat doesn’t flow from cool to hot. So, why does the model predict that energy does flow from cool to hot?

These two peculiarities cancel out each other mathematically. Energy is conserved and there is a net heat transfer rate, 0.47, from the Lit side to the Dark side.

So, at a level of overall effect, one can’t say that the net result is non-physical.

Yet, to me, it seems decidedly peculiar that the model seems to conceptually rely on energy flows which don’t match what one would expect to see based on the underlying heat transfer mechanisms.

**Peculiar Energy Partition Asymmetry**

W&M’s DAET model energy partitioning rule has a peculiar asymmetry to it.

Suppose the energy partition fraction is 𝛾 = 0.9. According to the model:

- Of the insolation absorbed by the surface on the Lit side, 90% of that energy flux will be transferred from the surface to the air. This suggests
*strong*thermal coupling between the surface and the air. - Of the energy that warm air brings to the Dark side, only 10% of that energy will be transferred from the air to the surface. This suggests
*weak*thermal coupling between the surface and the air.

This sort of coupling, in which there is a bias toward the energy always flowing to the same destination, regardless of where it comes from, is not typical of physical systems.

Consider a partially reflective mirror separating two rooms:

- If the reflectivity is high, then light will mostly stay in the room
*where it originated*. - If the transmissivity is high, then light will move between the rooms equally easily
*in both directions*.

The rule M&W have adopted is analogous to a magic mirror that somehow traps light on one side of it, regardless of which side the light originated on. Mirrors don’t work that way.

Heat transport, too, is usually symmetric (assuming you reverse the temperature difference). So, by default, I’d expect thermal coupling on Noonworld to be symmetric, just as coupling is symmetric with mirrors.

In the DAET model, it is difficult to relate what is happening to anything real, like the relative temperatures of the surface and the air.

Given that the model’s connection to physics is mostly non-existent, it’s hard to prove that the physics of the model is wrong.

Even so, it seems unlikely that the asymmetric behavior posited by the DAET energy partitioning rule could correspond to any real physics.

**No Enforcement of the Second Law of Thermodynamics**

The DAET energy partitioning rule guarantees that the First Law of Thermodynamics (energy conservation) will be honored.

That’s good, as far as it goes.

However, there is nothing about the DAET model which clearly ensures that the Second Law of Thermodynamics will be honored.

The Second Law of Thermodynamics requires that heat only flows from hot to cold, not cold to hot (unless something like a heat pump is involved).

For their model representing Venus, M&W calculated that the atmosphere of the planet is at 464℃. They also assumed that over 99% of insolation energy absorbed by the surface is transferred to the atmosphere. Yet, their model said the surface is at -46℃. So, the predicted heat transfer seems to involve heat flowing from cold to hot.

Of course, M&W’s calculation of that 464℃ temperature was not valid. So, maybe the model doesn’t really predict net heat transfer from cold to hot.

But the point is, it’s not clear that the DAET energy-partitioning rule prevents such Second-Law-violating outcomes from happening. Maybe it does. Maybe it doesn’t.

Without the Second Law as a constraint, it’s all too easy for a model to predict outcomes that are incompatible with reality.

**Game-World Physics**

Expecting a model like DAET to predict real-world physics is a little like expecting a computer game to accurately predict how reality functions.

M&W’s DAET model is based on an arbitrary energy-partitioning rule with no particular connection to the physics that M&W are trying to model.

When a model relies on arbitrary assumptions, the model can yield arbitrary outcomes. There is no reason to expect the results to have much to do with reality.

**Significance of the Model**

M&W have asserted that their DAET model explains how a planet like Venus can be much warmer than the “vacuum planet” temperature (computed by balancing absorbed insolation with thermal radiation emitted, for a planet of uniform temperature with the relevant albedo and surface emissivity).

Because M&W used an inappropriate formula for calculating atmospheric temperature, M&W’s assertion is false.

Properly interpreted, the DAET model does not predict or explain any atmospheric enhancement of temperature beyond the “vacuum planet” value.

The model does show that, given sufficiently powerful convective heat transfer, the temperatures of the lit and dark sides of a tidally locked planet could be nearly equalized. This is a valid conclusion, but it is not a surprising one.

**Conclusion**

I admire the passion, creativity, and effort that Philip Mulholland and Stephen Wilde have given to thinking freshly about how planetary climates function.

Regrettably, their DAET model doesn’t have much to tell us about real planets.

*[This essay was originally published as Trouble in Noonworld, Take 2.]*

- 1I didn’t realize at the time that that particular paper did not reveal the full essence of M&W’s approach to modeling. So, while my critique reflected what it was like to try to make sense of one of M&W’s papers in isolation, it didn’t address their full model.